Is complement $L = \{ w : |w|_{a} \equiv |w|_{b} \vee |w|_{c} \equiv |w|_{d} \}$ context-free

Question

$L = \{ w : |w|_{a} \equiv |w|_{b} \vee |w|_{c} \equiv |w|_{d} \}$

In my opinion complement of the L language is

$L^{C} = \{ w : |w|_{a} \neq |w|_{b} \wedge |w|_{c} \neq |w|_{d} \}$

I choose to Ogden pummping lemma word $s = a^{p}b^{p + p!}c^{p}d^{p + p!}$ and $ p > n$

I would like to distinguish $c ^ {p}$. And then I have to have at least one distinguished symbol and the rest not distinguished and in my opinion it can't be pumped in any case because i can $c^{p}$ pumped to $c^{p + p!}$ so it isn't context-free

Do I think right?